Applied Mathematics 3 Question Paper - Dec 18 - Computer Engineering (Semester 3)

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Applied Mathematics 3 - Dec 18

Computer Engineering (Semester 3)

Total marks: 80
Total time: 3 Hours INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.

1.a. If Laplace transform of ![\Large erf($\sqrt{t}$) = $\dfrac{1}{s \sqrt{s+1} }$, then find L{ $e^{t}$ .erf(2$\sqrt{t}$)}

(5 marks) 00

1.b. Find the Orthogonal Trajectory of the family of curves given by $e^{-x}$ .cos y + x.y = c

(5 marks) 00

1.c. Find Complex Form of Fourier Series for . $e^{2x}$ ; 0 < x < 2

(5 marks) 00

1.d. If the two regression equations are 5x - 6y + 90, 15x - 8y -180 = 0, find the means of x and y, the Correlation Coefficient and Standard deviation of x if variance of Y is 1

(5 marks) 00

2.a. Show that the function is Harmonic and find the Harmonic Conjugate v = $e^{x}$ .cos y + $x^{3}$ - 3x $y^{2}$

(6 marks) 00

2.b. Find Laplace Transform of enter image description here

(6 marks) 00

2.c. Find Fourier Series expansion of f(x) = x - $x^{2}$, -1 < x < 1

(8 marks) 00

3.a. Find the Analytics function f(z) = u + iv if v = log$\large \left ( x^{2} + y^{2} \right)$ + x - 2y

(6 marks) 00

3.b. Find Inverse Z Transform of enter image description here

(6 marks) 00

3.c. Solve the Differential Equation enter image description here

using Laplace Transform

(8 marks) 00

4.a. Find enter image description here

(6 marks) 00

4.b. FInd the spearman's Rank correlation coefficient between X and Y. | X | 60 | 30 | 37 | 30 | 42 | 37 | 55 | 45 |

| Y | 50 | 25 | 33 | 27 | 40 | 33 | 50 | 42 |

(6 marks) 00

4.c. Find Inverse Laplace transform of i) $\dfrac{3s + 1}{ (s+1)^{4} }$ ii) $\dfrac{e^{4-3s}}{ (s+4)^{5/2}}$

(8 marks) 00

5.a Find Inverse Laplace Transform using Convolution theorem $\dfrac{1}{ (s-4)^{2}(s+3)}$

(6 marks) 00

5.b. Show that the functions enter image description here

are Orthogonal on (-1,1). Determine the constants a,b such that functions f(x) = -1 + ax + b$x^{2}$ is Orthogonal to both enter image description here

on the (-1,1)

(6 marks) 00

5.c. Find the Laplace transform of enter image description here

(8 marks) 00

6.a. Fit a second degree parabola to the given data.

| X | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |

| Y | 1.1 | 1.3 | 1.6 | 2 | 2.7 | 3.4 | 4.1 |

(6 marks) 00

6.b Find the image of enter image description here

under the transformation w = $\dfrac{3 - z}{z - 2}$

(6 marks) 00

6.c Find Half Range Cosine Series for f(x) = xsinx in (0,$\small \pi$) and hence find $\dfrac{1}{1.3}$ - $\dfrac{1}{3.5}$ + $\dfrac{1}{5.7}$ - ....... = $\dfrac{\small \pi - 2}{4}$

(8 marks) 00

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